# Influence of Cross-Sectional Shape on Flow Capacity of Open Channels

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## Abstract

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_{Q}, was obtained. The advantage of our method is that the expression can be used to calculate the parameter directly, which varies with general and flow field characteristics. Another advantage is that the parameter can describe general and flow field characteristics in a uniform way. (2) The width-to-wetted perimeter ratio was selected to describe the cross-sectional shapes. The dependence of the parameter, C

_{Q}, on cross-sectional shapes can be summarized as follows: In laminar flow, the parameter depends on the width-to-wetted perimeter ratio only. In turbulent flow at a medium or low Reynolds number, the parameter varied with width-to-wetted perimeter ratio and Reynolds number. In turbulent flow at a high Reynolds number, the parameter was independent of the width-to-wetted perimeter ratio.

## 1. Introduction

^{3}/s); A is the area (m

^{2}); R is the hydraulic radius (m); S

_{f}is the energy slope (dimensionless); and C is the Chezy coefficient (dimensionless).

_{s}is the rough height (m); ${a}_{f}$ and ${b}_{f}$ are empirical coefficients [2] related to the discharge and operating conditions (dimensionless); φ is a coefficient related to the shape, e.g., for a rectangular section, φ = 0.95, for a trapezoidal section, φ = 0.80, and for an equilateral triangle section, φ = 1.25; R is the hydraulic radius (m); and ${Re}_{d}=\frac{4RU}{\nu}$, Re

_{d}is the Reynolds number with the hydraulic diameter as the characteristic length.

_{e}, is related to the wetted perimeter (Chow [4]; Prinos and Townsend [5]). Dracos and Hardegger [6] and French [7] used a weighted hydraulic radius to estimate flow discharge using the single channel method. For calculating the discharge of symmetrical flood plains, Cantero et al. [3] provided an expression (Equation (6)) using a flow area A with a certain weight. This expression is also related to mean flow velocity U, which is influenced by shape factors such as hydraulic radius, flood plain depth, and wetted perimeter (Wormleaton et al. [8]). Khatua et al. [9] provided expressions of two kinds of wetted perimeter coefficients. The expressions contain the power function of the flow area ratio. The expressions used to calculate the flow rate are complicated (see Cantero et al. [3], Equation (9)). The expressions of the kinetic energy correction coefficient (or Coriolis coefficient) α and momentum correction coefficient (or Boussinesq coefficient) β are given by Blalock and Sturm [10], Field et al. [11], and Chaudhry and Bhallamudi [12], respectively. Then, the specific energy function E, momentum function S, and flow profile h = h(x) are calculated. The Froude number of compound open channels was approximated by Blalock and Sturm [10], and is related to shape parameters such as top width, wetted perimeter, hydraulic radius of the subsection, etc. This is a bulk cross-sectional Froude number, rather than a local Froude number valid for different points in a section. The bulk cross-sectional Froude number can be used to further calculate the water depth and specific energy (Blalock and Sturm [10]; Blalock and Sturm [13]; and Costabile and Macchione [14]).

_{b}/U

_{surf}, where U

_{b}is the depth-averaged velocity and U

_{surf}is the local surface velocity), thus affecting the change in average velocity and discharge capacity. Harpold et al. [16] also observed through experiments that the variation in Reynolds number would affect the variation in the velocity index. The velocity index was set at 0.85 under base flow conditions and 0.93 under high flow conditions. Chen [17] and Schlichting [18] found that the logarithmic law of flow velocity is suitable for high Reynolds number flows, while the power law is valid for lower Reynolds number intervals. The coefficients of these functions are generally obtained experimentally.

_{Q}, which describes the influence of cross-sectional shape on flow capacity, through backward derivation, generalization, and simplification. The total flow energy equation (one of the governing equations of the integral model) established from the perspective of hydrodynamics solves the following shortcomings in the total flow energy equation in hydraulics: (1) The direct relationship between flow field description in fluid mechanics and total flow description in hydraulics cannot be built. (2) Distinguishing between total flow energy equations in laminar flow conditions and turbulent flow conditions in an open channel is impossible. (3) The effect of viscous dissipation and turbulence on total flow energy loss cannot be calculated directly, so the direct expression of total flow energy loss cannot be given. (4) The pressure on the cross-section must obey the static pressure distribution, but it is difficult to satisfy in the conditions of secondary flow and turbulent flow. This article is based on a new parameter, C

_{Q}, that can describe the effect of cross-sectional shapes on discharge capacity, and the following work was carried out: (1) The general expression of open-channel flow capacity was deduced firstly through theoretical analysis. At the same time, the expression of the flow capacity coefficient C

_{Q}was given. (2) On the basis of the general expression, the direct formula for calculating the coefficient of flow capacity, C

_{Q}, under laminar flow conditions was given. (3) Additionally, on the basis of the general expression, combined with the experiment results of this article and the literature, the variation trend of C

_{Q}under different turbulent flow conditions was obtained. (4) By means of numerical analysis, the variation trends of C

_{Q}in rectangular, trapezoidal, and compound open channels were obtained. (5) In the Discussion section, the variation trends of C

_{Q}in laminar and turbulent zones are shown in the same graph, using the experimental and numerical results. The differences in C

_{Q}in two regions are discussed. (6) The Conclusion section summarizes the variation trend of C

_{Q}in laminar, low and medium Reynolds number turbulent region, and high Reynolds number turbulent region. The research objective of this article was achieved; that is, we identified a flow capacity parameter C

_{Q}to represent the flow discharge capacity. The calculation formula is a direct expression, which is convenient for application in engineering calculations. At the same time, the width-to-wetted perimeter ratio was used to generalize the cross-sectional shape. The variation trend of C

_{Q}with width-to-wetted perimeter ratio and Reynolds number was obtained.

## 2. Materials and Methods

#### 2.1. Theoretical Analysis

_{W}in the region with volume V between two cross-sections is (derived from equations in [21,22,23,24]):

^{2}/s); g is the acceleration of gravity (m/s

^{2}); Q is the discharge (m

^{3}/s); and s

_{ij}, $\stackrel{-}{{s}_{ij}}$, and $-\stackrel{-}{{u}_{i}^{\prime}{u}_{j}^{\prime}}$ represent the velocity variation rate of laminar flow, the time-mean velocity variation rate of turbulent flow, and the Reynolds stress of open channels, respectively. In the condition of constant uniform flow in open channels, the corresponding energy loss coefficient λ can be calculated as follows [24]:

^{2}); and U is the mean velocity (m/s).

^{3}); $\chi $ is the wetted perimeter (m); and L is the distance (m).

_{f}is the energy slope (dimensionless).

_{Q}is the parameter reflecting the influence of the cross-sectional shape on discharge capacity except hydraulic radius (hydraulic radius was used as a parameter on the right side of Equation (7)). C

_{Q}is expressed as:

_{R}is the Reynolds number, with hydraulic radius as the characteristic length scale. Equation (8) has the following advantages: (1) it is a direct expression and can directly describe how the discharge capacity parameter C

_{Q}varies with the general characteristics (i.e., energy loss coefficient) or flow field characteristics (i.e., time-mean velocity and Reynolds stress). (2) The discharge capacity parameters obtained from the general characteristics and flow field characteristics are consistent. The discharge capacity parameters of laminar flow and turbulent flow in open channels are discussed below.

#### 2.2. Laminar Flow in an Open Channel

#### 2.3. Experiments for Turbulence in an Open Channel

_{Q}on turbulence in an open channel, we performed experiments using a high-precision, varying slope, rectangular smooth flume. The experiment was done in Changjiang River Scientific Research Institute, Wuhan City, China. The experimental device is shown in Figure 3. The flume is 28 m long, 56 cm wide, and 70 cm high. The actual picture and details of the flume are shown in Figure 4. The range of slope variation is −1–7‰. In this experiment, the bottom slope is 2.5‰ and 5‰. The slope changing system is shown in Figure 4. The frequency converter and needle water level gauge are shown in Figure 5. The flow velocity was measured by acoustic Doppler velocimetry (ADV). The actual picture of the ADV meter is shown in Figure 6. An ultrasonic automatic water level meter was used to measure the water depth. The instrument can synchronously measure the water level at many points, and it can output the results according to the specified format. Figure 7 shows a physical picture of the instrument. In the experiment, seven high-precision ultrasonic water level probes were arranged along the flume (measuring Section 1#–7#, as shown in Figure 3). The installation height was no less than 5 cm from the water surface. The sampling frequency was 10 Hz. The experimental conditions are shown in Table 1. In the experiment, we mainly measured velocity and water depth. The sample size of the data collected via ADV was 1800 PCS/min. Each measuring point was sampled for 1 min, and the sampling signal frequency was 30 signals per second. Table 1 is the experiment conditions.

## 3. Results

#### 3.1. Laminar Flow in an Open Channel

_{Q}with width-to-wetted perimeter ratio B/χ. The numerical value is the result calculated using Equation (8) (or Equation (10)). The “laminar” expression in Equation (8) (or Equation (10)) and Figure 6 show that C

_{Q}is independent of the Reynolds number in laminar flow but strongly dependent only on B/χ. With the increase in B/χ, C

_{Q}gradually decreased. The numerical results in Figure 8 were fitted to derive an approximate calculation formula shown in Equation (11). The fitting results are shown in Figure 8.

#### 3.2. Experiments for Turbulence in an Open Channel

_{Q}, defined in Equation (8), was calculated. Then, the C

_{Q}~Re

_{R}~B/χ diagram was drawn, as shown in Figure 9. For rectangular open channels, C

_{Q}decreased with the increase in Reynolds number Re

_{R}. In addition, for turbulent flow at medium and low Reynolds numbers (Reynolds number < 10

^{5}) in different B/χ conditions, C

_{Q}were quite dispersed, showing that C

_{Q}is clearly dependent on B/χ, i.e., when B/χ is different, the corresponding C

_{Q}is also significantly different. However, after the Reynolds number reaches 10

^{5}, in different B/χ conditions, C

_{Q}is gradually concentrated into a line and the dependence of C

_{Q}on B/χ is weakened. C

_{Q}values of different B/χ values tended to overlap. C

_{Q}showed a trend only related to the Reynolds number.

#### 3.3. Numerical Calculations

#### 3.3.1. Mathematical Model and Verification

#### 3.3.2. Rectangular Open Channel

_{Q}with Reynolds number and width-to-wetted perimeter ratio in a turbulent rectangular open channel. To facilitate comparison, the corresponding experimental results with high Reynolds number turbulent flow conditions are also shown. Figure 11 shows that, similar to the experimental results in Figure 9, C

_{Q}decreased with increasing Reynolds number. In addition, for numerical values, when the Reynolds number is less than 10

^{5}, C

_{Q}is dispersed in different B/χ conditions, indicating that C

_{Q}still depends on B/χ. If the Reynolds number is greater than 10

^{5}, C

_{Q}is concentrated and it is assumed that C

_{Q}no longer depends on B/χ. In a high Reynolds number region (Reynolds number greater than 10

^{5}), the variation trend of experimental results and numerical results are consistent, which shows that C

_{Q}can reflect the variation trend of the flow discharge capacity. However, the experimental values were slightly smaller than the numerical values. A possible reason for this is that the actual energy loss was not fully taken into account in the numerical calculation. Therefore, the numerical C

_{Q}values were slightly higher.

#### 3.3.3. Open Channels with Other Cross-sections

_{Q}showed a similar trend to that of rectangular open channels: the C

_{Q}value decreased with increasing Reynolds number and C

_{Q}values were bounded by a Reynolds number of 10

^{5}, and C

_{Q}showed strong dependence on B/χ for turbulent flow at low and medium Reynolds numbers. However, in the case of high Reynolds number turbulent flow, C

_{Q}was no longer dependent on B/χ, but on the Reynolds number.

## 4. Discussion

_{Q}in conditions with various cross-sectional shapes (rectangular, trapezoidal, and compound), the numerical results and experimental results for turbulent flow and theoretical solutions for laminar flow were combined into the same figure (all for open-channel flow). Figure 15 and Figure 16 show the variation in C

_{Q}with Re

_{R}at a given width-to-wetted perimeter ratio (B/χ is 0.7 and 0.9, respectively). Figure 15 and Figure 16 show the following: (1) For medium and low Reynolds number turbulence with a Reynolds number less than 10

^{5}with the same B/χ value, the C

_{Q}of the three types of cross-sectional shapes (namely, rectangular, trapezoidal, and compound) were different from each other. The discharge capacity parameter C

_{Q}varied with the shape of the cross-section, and the mechanism was extremely complex; further study is needed. (2) For turbulent flow with a high Reynolds number greater than 10

^{5}, the numerical values of C

_{Q}in the three cross-sectional conditions were almost the same as the experimental values, and C

_{Q}was no longer dependent on the width-to-wetted perimeter ratio B/χ. In high Reynolds number turbulent flow conditions, turbulence structures at different scales were fully developed, and the discharge capacity parameter, C

_{Q}, was no longer dependent on the shape of the cross-section. The experimental and numerical results in Figure 15 and Figure 16 are mostly consistent. However, the experimental values in Figure 16 were slightly lower than the numerical results. A possible reason for this is that the actual energy loss was not fully taken into account in the numerical calculations. Therefore, the numerical C

_{Q}values were slightly higher.

## 5. Conclusions

_{Q}affecting the discharge capacity of laminar flow and turbulent flow in open channels with different types of cross-sections was obtained.

_{Q}was only related to the width-to-wetted perimeter ratio. For medium and low Reynolds number turbulence, parameter C

_{Q}varied with the Reynolds number and width-to-wetted perimeter ratio. However, for high Reynolds number turbulent flow, the discharge capacity parameter C

_{Q}did not vary with width-to-wetted perimeter ratio.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Flume in the experiment. (

**a**) Flume side view; (

**b**) top view of flume; (

**c**) inflow system; and (

**d**) slope changing system.

**Figure 5.**Flume in the experiment. (

**a**) Frequency converter (used for controlling the power of the inflow pump); (

**b**) needle water level gauge.

**Figure 8.**C

_{Q}vs. B/χ for laminar flow in rectangular open channels (data from numerical calculation).

**Figure 9.**C

_{Q}vs. Re

_{R}for rectangular open channels with different B/χ values (data from experiments).

**Figure 10.**Velocity profiles of cross-sections in experimental condition 3 (data from experiment and numeric calculations): (

**a**) 2# cross-section, (

**b**) 3# cross-section, (

**c**) 4# cross-section, and (

**d**) 5# cross-section.

**Figure 11.**C

_{Q}vs. Re

_{R}for rectangular open channels with different B/χ in turbulent flow (data from numerical calculation and Tracy et al. [19]).

**Figure 12.**Sketch of open channels with different cross-sections: (

**a**) trapezoidal open channel, (

**b**) compound open channel.

**Figure 13.**C

_{Q}vs. Re

_{R}for trapezoidal open channels with different B/χ (data from numerical calculation).

**Figure 14.**C

_{Q}vs. Re

_{R}for compound open channels with different B/χ (data from numerical calculation).

Case | Discharge Q (L/s) | Slop S_{f} | Water Depth H (cm) |
---|---|---|---|

1 | 20 | 0.0025 | 5.99 |

2 | 40 | 0.0025 | 9.41 |

3 | 60 | 0.0025 | 12.30 |

4 | 20 | 0.005 | 4.48 |

5 | 40 | 0.005 | 6.86 |

6 | 60 | 0.005 | 9.06 |

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**MDPI and ACS Style**

Qiu, C.; Liu, S.; Huang, J.; Pan, W.; Xu, R.
Influence of Cross-Sectional Shape on Flow Capacity of Open Channels. *Water* **2023**, *15*, 1877.
https://doi.org/10.3390/w15101877

**AMA Style**

Qiu C, Liu S, Huang J, Pan W, Xu R.
Influence of Cross-Sectional Shape on Flow Capacity of Open Channels. *Water*. 2023; 15(10):1877.
https://doi.org/10.3390/w15101877

**Chicago/Turabian Style**

Qiu, Chunlin, Shihe Liu, Jiesheng Huang, Wenhao Pan, and Rui Xu.
2023. "Influence of Cross-Sectional Shape on Flow Capacity of Open Channels" *Water* 15, no. 10: 1877.
https://doi.org/10.3390/w15101877